Radially Symmetric Solution for Fractional Laplacian Systems with Different Negative Powers

Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia

School of Mathematics and Computer Science, Shanxi Normal University, Taiyuan 030031, China

Authors to whom correspondence should be addressed.

Academic Editor: Palle Jorgensen

Fractal Fract. 2022, 6(7), 352; https://doi.org/10.3390/fractalfract6070352

Received: 26 May 2022 / Revised: 15 June 2022 / Accepted: 16 June 2022 / Published: 23 June 2022

(This article belongs to the Special Issue Initial and Boundary Value Problems for Differential Equations)

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Abstract

By developing the direct method of moving planes, we study the radial symmetry of nonnegative solutions for a fractional Laplacian system with different negative powers:

{(- Δ)}^{\frac{α}{2}} u (x) + u^{- γ} (x) + v^{- q} (x) = 0, x \in R^{N},

{(- Δ)}^{\frac{β}{2}} v (x) + v^{- σ} (x) + u^{- p} (x) = 0, x \in R^{N},

u (x) ≳ {| x |}^{a}, v (x) ≳ {| x |}^{b} as | x | \to \infty,

where

α, β \in (0, 2)

, and

a, b > 0

are constants. We study the decay at infinity and narrow region principle for the fractional Laplacian system with different negative powers. The same results hold for nonlinear Hénon-type fractional Laplacian systems with different negative powers. View Full-Text

Keywords: fractional Laplacian systems; different negative powers; radial symmetry; the direct method of moving planes

This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

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MDPI and ACS Style

Xu, H.; Ahmad, B.; Wang, G.; Zhang, L. Radially Symmetric Solution for Fractional Laplacian Systems with Different Negative Powers. Fractal Fract. 2022, 6, 352. https://doi.org/10.3390/fractalfract6070352

AMA Style

Xu H, Ahmad B, Wang G, Zhang L. Radially Symmetric Solution for Fractional Laplacian Systems with Different Negative Powers. Fractal and Fractional. 2022; 6(7):352. https://doi.org/10.3390/fractalfract6070352

Chicago/Turabian Style

Xu, Haiyong, Bashir Ahmad, Guotao Wang, and Lihong Zhang. 2022. "Radially Symmetric Solution for Fractional Laplacian Systems with Different Negative Powers" Fractal and Fractional 6, no. 7: 352. https://doi.org/10.3390/fractalfract6070352

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Radially Symmetric Solution for Fractional Laplacian Systems with Different Negative Powers

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